nLab
multinomial coefficient

Context

Arithmetic

Combinatorics

Contents

Idea

For n 0n \in \mathbb{N}_0 a natural number or zero and (k i 0) i=1 r(k_i \in \mathbb{N}_0)_{i = 1}^r with ik i=k\underset{i}{\sum} k_i = k, the corresponding multinomial coefficient

(nk 1,k 2,,k r)n!k 1!k 2!k r \left( n \atop { k_1, k_2, \cdots, k_r } \right) \;\coloneqq\; \frac{ n! }{ k_1 ! \, k_2 ! \, \cdots \, k_r } \;\in\; \mathbb{N}

is the quotient of the factorial of nn by the multiplication of the factorials of the k ik_i.

This number is the number of ways of drawing k 1k_1 elements and then k 2k_2 elements and so forth from nn elements, all in an unordered way.

For r=2r = 2 this is the binomial coefficient.

References

See also

Last revised on August 26, 2018 at 08:15:42. See the history of this page for a list of all contributions to it.